Duality, Descent & Defects The Higher Geometry and Category Theory of Charged Dynamics (II)

Lecturer: Rafał R. Suszek

Mode: hybrid

Time: Fridays, 1715—1900, lecture room 1.40

Basis for crediting: an oral exam or a report upon completion of a self-study project.

Self-study projects:

The Cartan–Borel construction — from the homotopy model of the orbispace through equivariant cohomology to equivariant structures on n-gerbes.
Current symmetries in the WZNWG σ-model with defects and their higher-geometric lift.
The Hopf index — the fibre-bundle and cohomology perspectives.
Elements of the spectral non-commutative geometry à la Connes, and its gauge field-theoretic deformations.
The gerbe theory of loop dualities (e.g., the T-duality).
Quantum groups as symmetry models in the two-dimensional Rational Conformal Field Theory.
Cohomology descent to shadow bundles of principaloid bundles with Lie-groupoid symmetry models.

Lecture notes:

Literature:

Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, 2005.

Pedro Martínez Gadea & Jaime Muñoz Masqué, Analysis and Algebra on Differentiable Manifolds. A Workbook for Students and Teachers, Springer, 2009.

Shoshichi Kobayashi & Katsumi Nomizu, Foundations of Differential Geometry, vols. 1-2, John Wiley & Sons, 1963-69.

Peter W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, Vol. 93, American Mathematical Society, 2008.

Ivan Kolář, Peter W. Michor & Jan Slovák, Natural Operations in Differential Geometry, Springer, 1993.

Peter W. Michor, Gauge Theory for Fiber Bundles, Extended version of a series of lectures held at the Institute of Physics of the University of Napoli, March 28—April 1, 1988.

Yvonne Choquet-Bruhat, Cécile DeWitt-Morette & Margaret Dillard Bleick, Analysis, Manifolds and Physics, Part I, II, Elsevier, 1996-2000.

John M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Vol. 218, Springer, 2013.

Борис А. Дубровин, Сергей П. Новиков, Анатолий Т. Фоменко, Modern Geometry - Methods and Applications, vol. 1,2,3, Graduate Texts in Mathematics, Vol. 93, 104, 124, Springer, 1984-90 (a translation of the original work Современная геометрия, Методы и приложения, Том 1, 2, 3, Наука, 1979).

Raoul Bott & Loring W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Vol. 82, Springer, 1982.

Nicolas Bourbaki, Éléments de mathématique. Livre II: Algèbre, chapitres 1 à 3, Springer, 2007.

Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics Vol. 5, Springer, 1971.

Tom Leinster, Basic Category Theory, Cambridge Studies in Advanced Mathematics Vol. 143, Cambridge University Press, 2014.

Claude Chevalley, The Algebraic Theory of Spinors and Clifford Algebras, Collected Works of Claude Chevalley, Vol. 2, Springer-Verlag, 1996.

Pertti Lounesto, Clifford Algebras and Spinors, London Mathematical Society Lecture Notes Series, Vol. 286, Cambridge University Press, 2001.

É. Cartan, The Theory of Spinors, Dover Books on Mathematics, Dover Publications, 1981 (tłumaczenie oryginalnych notatek wykładowych ''Leçons sur la théorie des spineurs'' zebranych w 1937 r. przez A. Merciera).

H. Blaine Lawson i Marie-Louise Michelsohn, Spin Geometry, Princeton Mathematical Series, Vol. 38, Princeton University Press, 1989.

Paul Budinich i Andrzej M. Trautman, The Spinorial Chessboard, Trieste Notes in Physics, Springer-Verlag, 1988.

Stephen A. Huggett i K. Paul Tod, An Introduction to Twistor Theory, London Mathematical Society Student Texts, Vol. 4, Cambridge University Press, 1985.

Loring W. Tu, Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies, Vol. 204, Princeton University Press, 2020.

Complementary material:

R.R.S., „Algebra. Podstawy”, skrypt do wykładu z „Algebry I i II R” (wersja przed-ostateczna, wszelkie uwagi mile widziane).

R.R.S., „Niezbędnik różniczkowo–geometryczny” (wersja przed-ostateczna, wszelkie uwagi mile widziane).

R.R.S., ,,Action directe grupy Liego'' — ucieczka z Kina Powszedniość w przestrzenie jednorodne przy użyciu Twierdzenia Cartana (o podgrupie domkniętej) i Twierdzenia o rozmaitości ilorazowej

R.R.S., Lecture notes from the monographic course ,,Methods of Higher Algebra in Physics I & II — from quadratic forms to spinor bundles''

R.R.S., Minicourse on ''Gauge symmetries and their higher analogues'' at the Bucharest 2024 Minischool ''Mathematical methods in Gravitation and Cosmology'', November 13-17, 2024, București–Magurele, România: Abstract, Lecture I, Lecture II, Lecture III, Tutorial

Monographic lecture course Duality, Descent & Defects The Higher Geometry and Category Theory of Charged Dynamics (II)Academic year 2024/2025, summer semester